Discrete phase space - III: The divergence-free S-matrix elements

Abstract

In the arena of the discrete phase space and continuous time, the theory of S-matrix is formulated. In the special case of quantum electrodynamics (QED), the Feynman rules are precisely developed. These rules in the four-momentum turn out to be identical to the usual QED, except for the vertex function. The new vertex function is given by an infinite series that can only be treated in an asymptotic approximation at the present time. Preliminary approximations prove that the second-order self-energies of a fermion and a photon in the discrete model have convergent improper integrals. In the final section, a sharper asymptotic analysis is employed. It is proved that in case where the number of external photon or fermion lines is at least one, then the S-matrix elements converge in all orders. Moreover, there are no infrared divergences in this formulation.